Answer to Question #266117 in Analytic Geometry for Almas

Question

Definition, Formula and Physical interpretation of Cross Product.

Answer 1

The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

The cross product is defined by the formula

$$\vec a\times\vec b=||\vec a||\cdot||\vec b||\sin \theta \vec n$$

where:

$θ$  is the angle between $\vec a$  and $\vec b$  in the plane containing them (hence, it is between 0° and 180°)

$||\vec a||$ and $||\vec b||$ are the magnitudes of vectors $\vec a$ and $\vec b$

and $\vec n$  is a unit vector perpendicular to the plane containing  $\vec a$ and $\vec b$ , in the direction given by the right-hand rule .

If the vectors  $\vec a$ and $\vec b$  are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of $\vec a$ and $\vec b$  is the zero vector  $\vec 0.$

If $(i, j,k)$ is a positively oriented orthonormal basis, the

$$\vec a\times\vec b=\begin{vmatrix} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}$$

The cross product is used to describe the Lorentz force experienced by a moving electric charge $q_e:$

$$\vec F=q_e(\vec E+\vec v\times \vec B)$$

Since velocity $\vec v,$  force $\vec F$  and electric field $\vec E$  are all true vectors, the magnetic field $\vec B$  is a pseudovector.

The moment $\vec M$  of a force $\vec F_B$  applied at point B around point A is given as:

$$\vec M_A=\vec r_{AB}\times\vec F_B$$

The angular momentum $\vec L$  of a particle about a given origin is defined as:

$$\vec L=\vec r\times \vec p,$$

where $\vec r$  is the position vector of the particle relative to the origin, $\vec p$   is the linear momentum of the particle.